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Math 1304 Calculus I

Math 1304 Calculus I. 2.6 – Limits involving Infinity, Asymptotes. Recall: Limits of Infinity. Recall: Vertical Asymptotes. Definition: The line x = a is called a vertical asymptote of the curve y = f(x) if any of the following limits exist. Limits as x approaches ± infinity. Now consider:.

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Math 1304 Calculus I

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  1. Math 1304 Calculus I 2.6 – Limits involving Infinity, Asymptotes

  2. Recall: Limits of Infinity

  3. Recall: Vertical Asymptotes • Definition: The line x = a is called a vertical asymptote of the curve y = f(x) if any of the following limits exist

  4. Limits as x approaches ±infinity Now consider:

  5. Basic Examples • f(x) = 1/x • f(x) = 1/(x-a) • f(x) = 1/x2 • f(x)= 1/(x-a)2

  6. Def of Limit as x approaches infinity • Definition: Let f be a function defined on some interval (a,). We say that the limit as x approaches infinity is L if for any >0 there is a number N such that |f(x)-L|<  whenever x > N. In this case we write

  7. Def of Limit as x approaches -infinity • Definition: Let f be a function defined on some interval (-,a). We say that the limit as x approaches minus infinity is L if for any >0 there is a number N such that |f(x)-L|<  whenever x < N. In this case we write

  8. Def of Limit of infinity as x approaches infinity • Definition: Let f be a function defined on some interval (a,). We say that the limit as x approaches infinity is infinity if for any M there is a number N(M) such that f(x)>M whenever x > N(M). In this case we write

  9. Horizontal Asymptotes • Definition: The line y = L is called a horizontal asymptote of the curve y = f(x) if either of the following limits exist

  10. Computational Methodsfor limits as x approaches +-infinity • Rules • Basic functions: constants, 1/x, 1/xr • Sum, difference, product, quotient, power, etc. • Algebraic Techniques • For quotient of polynomials, divide by highest power (example next)

  11. Basic Functions

  12. Examples • Find all horizontal and vertical asymptotes of the curve y=(2x-1)/(5x+3) • Find all horizontal and vertical asymptotes of the curve y=(2x2-1)/(3x2+3x+6)

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